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A120924 Triangle read by rows: T(n,k) is the number of ternary words of length n on {0,1,2}, having k isolated 0's (n >= 0, k >= 0).

%I A120924 #7 Nov 26 2019 04:40:20
%S A120924 1,2,1,5,4,13,12,2,33,36,12,83,108,48,4,209,316,172,32,527,904,588,
%T A120924 160,8,1329,2548,1932,672,80,3351,7104,6140,2592,480,16,8449,19628,
%U A120924 19020,9440,2320,192,21303,53816,57756,32896,10000,1344,32,53713,146596
%N A120924 Triangle read by rows: T(n,k) is the number of ternary words of length n on {0,1,2}, having k isolated 0's (n >= 0, k >= 0).
%C A120924 Row n has 1+ceiling(n/2) terms.
%C A120924 Row sums are the powers of 3 (A000244).
%C A120924 T(n,0) = A120925(n).
%C A120924 Sum_{k=0..ceiling(n/2)} k*T(n,k) = A120926(n).
%F A120924 G.f. = G(t,z) = (1-(1-t)z(1-z))/(1 - 3z + 2(1-t)z^2*(1-z)).
%e A120924 T(2,0)=5 because we have 00,11,12,21 and 22; T(2,1)=4 because we have 01,02,10 and 20; T(3,2)=2 because we have 010 and 020.
%e A120924 Triangle starts:
%e A120924    1;
%e A120924    2,   1;
%e A120924    5,   4;
%e A120924   13,  12,  2;
%e A120924   33,  36, 12;
%e A120924   83, 108, 48, 4;
%p A120924 G:=(1-(1-t)*z*(1-z))/(1-3*z+2*(1-t)*z^2*(1-z)): Gser:=simplify(series(G,z=0,15)): P[0]:=1: for n from 1 to 13 do P[n]:=sort(coeff(Gser,z^n)) od: for n from 0 to 13 do seq(coeff(P[n],t,j),j=0..ceil(n/2)) od; # yields sequence in triangular form
%Y A120924 Cf. A000244, A105114, A120925, A120926.
%K A120924 nonn,tabf
%O A120924 0,2
%A A120924 _Emeric Deutsch_, Jul 16 2006